Subgroups — #27367939

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284516	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20274170	C₂	1	4	yes	maximal, complemented
4	#20274116	C₂	1	2	yes	maximal, complemented
5	#20274082	C₂	1	2	yes	maximal, complemented
6	#20273474	C₂	1	1	yes	maximal, characteristic
7	#20273405	C₂	1	2	yes	maximal
8	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16810927	—	2	4	no	—
11	#16810915	C₂²	1	4	yes	—
12	#16810905	C₂²	1	4	yes	—
13	C₈×Dic₁₀₆	—	2	1	no	—
14	C₈×Dic₁₀₆	—	2	1	no	—
15	C₈×Dic₁₀₆	C₂²	1	2	yes	—
16	C₈×Dic₁₀₆	C₂²	1	2	yes	—
17	#16810666	—	2	2	no	—
18	#16810647	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×#13133146	—	2	1	no	—
21	#16810631	—	2	2	no	—
22	C₂×C₈×Dic₅₃	—	2	1	no	—
23	#16810376	C₂²	1	2	yes	—
24	#16810376	C₂²	1	2	yes	—
25	#16810372	C₂²	1	4	yes	—
26	#16810369	C₂²	1	4	yes	—
27	C₅₃×#533	C₂²	1	2	yes	—
28	C₅₃×#500	C₂²	1	1	yes	characteristic
29	C₅₃×#494	C₂²	1	1	yes	characteristic
30	C₅₃×#487	C₂²	1	2	yes	—
31	C₅₃×#467	—	2	1	no	—
32	C₅₃×#467	—	2	1	no	—
33	C₄×Q₁₆×C₅₃	—	2	1	no	—
34	C₄×Q₁₆×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
37	C₂×#11719917	—	2	2	no	—
38	C₂²×#88322	—	2	1	no	—
39	C₄×Dic₁₀₆	—	2	1	no	—
40	C₄×Dic₁₀₆	—	2	1	no	—
41	C₄×Dic₁₀₆	D₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
44	C₂×#11719919	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	#13133150	—	2	4	no	—
47	#13133148	—	2	4	no	—
48	C₄⋊#88322	—	2	1	no	—
49	C₄⋊#88322	—	2	2	no	—
50	C₄⋊#88322	C₂³	1	2	yes	—
51	C₄⋊#88322	C₂³	1	4	yes	—
52	#13133146	—	2	2	no	—
53	#13133146	—	2	2	no	—
54	#13133140	—	2	4	no	—
55	#13133140	—	2	4	no	—
56	C₈×Dic₅₃	—	2	2	no	—
57	C₈×Dic₅₃	—	2	2	no	—
58	C₂×Q₁₆×C₅₃	—	2	1	no	—
59	C₂×Q₁₆×C₅₃	—	2	1	no	—
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
64	#125×C₅₃	—	2	2	no	—
65	C₂²⋊Q₈×C₅₃	—	2	2	no	—
66	C₂²⋊Q₈×C₅₃	—	2	2	no	—
67	C₄×Q₈×C₅₃	—	2	2	no	—
68	C₄×Q₈×C₅₃	—	2	2	no	—
69	#117×C₅₃	—	2	2	no	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	—	2	1	no	—
72	#107×C₅₃	—	2	1	no	—
73	#103×C₅₃	—	2	2	no	—
74	#103×C₅₃	—	2	2	no	—
75	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	2	2	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂²×Dic₅₃	—	2	1	no	—
85	C₂²×Dic₅₃	—	2	1	no	—
86	C₂×Dic₁₀₆	—	2	1	no	—
87	C₂×Dic₁₀₆	—	2	2	no	—
88	C₂×Dic₁₀₆	Q₁₆	1	4	yes	complemented
89	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
90	Q₁₆×C₅₃	—	4	2	no	—
91	Q₁₆×C₅₃	—	4	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₂²⋊C₄×C₅₃	—	4	2	no	—
105	C₄²×C₅₃	—	4	2	no	—
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	#41	1	1	yes	characteristic
109	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₄⋊Dic₅₃	—	2	1	no	—
111	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
112	#11719919	—	2	2	no	—
113	#11719919	—	2	2	no	—
114	C₄×Dic₅₃	—	2	2	no	—
115	C₄×Dic₅₃	—	2	2	no	—
116	#11719917	—	4	4	no	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	—	4	2	no	—
120	C₂³×C₅₃	#121	1	1	yes	characteristic
121	Q₈×C₅₃	—	4	2	no	—
122	Q₈×C₅₃	—	4	2	no	—
123	Q₈×C₅₃	—	8	2	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄×C₅₃	#135	1	2	yes	—
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₅₃	—	2	2	no	—
138	C₂×Dic₅₃	—	2	4	no	—
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	4	2	no	—
141	Dic₁₀₆	—	2	2	no	—
142	Dic₁₀₆	—	2	4	no	—
143	Dic₁₀₆	C₂×Q₁₆	1	4	yes	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	—	2	1	no	—
147	C₈×C₅₃	#137	1	2	yes	—
148	#88322	—	4	4	no	—
149	#88322	—	4	2	no	—
150	#19262	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	#467	1	2	yes	—
154	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	#467	1	2	yes	—
161	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
162	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
163	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
164	C₄×C₅₃	#575	1	1	yes	characteristic
165	Dic₅₃	—	2	4	no	—
166	Dic₅₃	—	4	2	no	—
167	#3170	D₅₃	1	1	yes	characteristic
168	#3078	—	53	2	no	—
169	C₂×C₈×Q₈	—	53	1	no	—
170	#1684	—	53	4	no	—
171	#1676	—	53	2	no	—
172	#1670	—	53	2	no	—
173	#1567	—	53	1	no	—
174	#1550	—	53	2	no	—
175	C₂×C₅₃	—	2	2	no	—
176	C₂×C₅₃	#3037	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
178	C₂×C₅₃	#3067	1	1	yes	characteristic
179	#533	C₂×D₅₃	1	2	yes	—
180	C₂×C₄×Q₈	—	53	1	no	—
181	#500	C₂×D₅₃	1	1	yes	characteristic
182	#494	C₂×D₅₃	1	1	yes	characteristic
183	#487	C₂×D₅₃	1	2	yes	—
184	#479	—	106	4	no	—
185	#477	—	53	4	no	—
186	#475	—	53	4	no	—
187	#467	—	2	1	no	—
188	#467	—	2	1	no	—
189	C₈×Q₈	—	53	2	no	—
190	C₈×Q₈	—	53	2	no	—
191	C₈×Q₈	—	106	1	no	—
192	C₈×Q₈	—	106	1	no	—
193	C₄×Q₁₆	—	2	1	no	—
194	C₄×Q₁₆	—	2	1	no	—
195	#428	—	106	2	no	—
196	#423	—	53	2	no	—
197	C₂×C₄⋊C₈	—	53	1	no	—
198	C₂×C₄⋊C₈	—	106	1	no	—
199	#419	—	106	2	no	—
200	C₂×C₄×C₈	—	106	1	no	—
201	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
202	#335	—	53	2	no	—
203	#335	—	53	2	no	—
204	#333	—	53	4	no	—
205	#332	—	53	4	no	—
206	C₅₃	#19262	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	—	53	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×M₁₆	—	106	2	no	—
211	C₂²×C₈	—	106	1	no	—
212	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
215	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
216	#125	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	53	2	no	—
222	C₄×Q₈	—	53	2	no	—
223	C₄×Q₈	—	106	1	no	—
224	C₄×Q₈	—	106	1	no	—
225	#117	—	2	2	no	—
226	C₂×C₄⋊C₄	—	53	1	no	—
227	C₂×C₄⋊C₄	—	106	1	no	—
228	C₂×C₄²	—	106	1	no	—
229	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	—	106	4	no	—
233	#106	—	106	4	no	—
234	C₄⋊C₈	—	53	2	no	—
235	C₄⋊C₈	—	53	4	no	—
236	C₄⋊C₈	—	106	2	no	—
237	C₄⋊C₈	—	106	2	no	—
238	C₄⋊C₈	—	106	2	no	—
239	C₄⋊C₈	—	106	1	no	—
240	#103	—	2	2	no	—
241	#103	—	2	2	no	—
242	#103	—	106	4	no	—
243	#103	—	106	4	no	—
244	C₄×C₈	—	106	2	no	—
245	C₄×C₈	—	106	2	no	—
246	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	53	4	no	—
253	C₂×Q₈	—	53	2	no	—
254	C₂×Q₈	—	106	1	no	—
255	C₂×Q₈	—	106	2	no	—
256	C₂²×C₄	—	106	1	no	—
257	C₂²×C₄	—	106	1	no	—
258	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
259	C₂²×C₄	#11719947	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
261	Q₁₆	—	4	2	no	—
262	Q₁₆	—	4	2	no	—
263	M₁₆	—	212	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	106	4	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	212	2	no	—
269	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	53	2	no	—
280	C₄⋊C₄	—	106	1	no	—
281	C₄⋊C₄	—	106	2	no	—
282	C₄⋊C₄	—	106	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	—	106	2	no	—
286	C₄²	—	106	2	no	—
287	C₄²	D₄×D₅₃	1	1	yes	characteristic
288	C₄²	#11719947	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
290	C₄²	C₂³×D₅₃	1	1	yes	characteristic
291	C₂³	#13133188	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	53	4	no	—
296	Q₈	—	106	4	no	—
297	Q₈	—	106	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	4	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	#13133188	1	1	yes	characteristic
308	C₂×C₄	#13133188	1	1	yes	characteristic
309	C₂×C₄	#13133188	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
311	C₂×C₄	#13133240	1	2	yes	—
312	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	212	4	no	—
319	C₈	—	212	2	no	—
320	C₈	#13133247	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#16810861	1	2	yes	—
324	C₂²	C₂×#13133188	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	106	4	no	—
331	C₄	—	212	2	no	—
332	C₄	#16810861	1	2	yes	—
333	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
334	C₄	C₂×#13133240	1	1	yes	characteristic
335	C₄	C₂×#13133247	1	1	yes	characteristic
336	C₄	#16811421	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#20284183	1	1	yes	characteristic
339	C₂	C₂×#16810861	1	1	yes	characteristic
340	C₂	#20284431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×C₅₃	#121
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#19262
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20274170	C₂	semidirect
4. #20274116	C₂	semidirect
5. #20274082	C₂	semidirect
88. C₂×Dic₁₀₆	Q₁₆	semidirect
206. C₅₃	#19262	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20284516	2
C₂×C₈×Dic₁₀₆	2
#20274170	2
#20274116	2
#20274082	2
#20273474	2
#20273405	2
C₅₃×#3170	2
#19262	53